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Theory?

I offer this to my students, so I guess it can't hurt to offer here, so here goes! If you have any questions about theory, let me know via a post here etc and I'll try to explain in way that makes sense. Okay?

Music education opens doors that help children pass from school into the world around them-a world of work, culture, intellectual activity, and human involvement. The future of our Nation depends on providing our children with a complete education that includes music. -Gerald Ford

Lucky for you, I've provided this for you! Okay let's get started. First of all, what is transposition? Wikipedia defines it as such:

The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.

Make sense? No? When a Bb Trumpet plays a concert Bb majour scale, they are playing their C majour scale. When an Eb saxaphone plays a concert Eb majour scale, they are playing their scale. So let's redefine transposition:

Taking the same series of steps so that it sounds the same on an instrument of a different key.

Okay at this point, let's take a look at our Circle of Fifths. Well firstly why do we call it a circle and not say a line? Because no matter where you start, you can go clockwise or counter clockwise completely around the circle and just keep going and going. This makes playing scales for instruments of different keys much much easier. If you'll notice, around the circle I gave you, each scale says something like "1b", "3#", etc. If we start at concert C, and number each one, it should look like this:
Your sharp keys are in red and positive, and your flat keys are in blue are negative. 0 and +-6 are in orange.

Let's say your instrument is in Bb. So on the circle we should try to rotate the circle so that 0 is at Bb:
Now the Bb scale is C (0 sharps or flats). Now let's say your conductor/teacher/band director/etc asks you to play a concert C majour scale. Well according to our Circle of Fifths that has been rotated for Bb, a concert C majour scale has 2 sharps.

That's great, what does that have to do anything with anything? All music is based on how the notes relate together. All music can be broken down into the basic relationship between notes. What I mean by this is in your concert Bb majour scale, you play:
Bb C D Eb F G A Bb
This is a predetermined pattern of whole whole half whole whole whole half that all majour scales will always follow this pattern.

Well that's great, but how do I apply this to a melody? That's simple too! Let's take Twinkle Twinkle Little Star for an example:
This version happens to be in C, our base note is C. I've added the scale break down here (root, fifth, etc):
What does this mean exactly? Well, anything that follows that pattern of 00445543322110 will sound like twinkle twinkle little star.
Section A and Section B are written in their respective key (C, Bb), however, the melody is still twinkle twinkle little star!

Hope this helps!

Music education opens doors that help children pass from school into the world around them-a world of work, culture, intellectual activity, and human involvement. The future of our Nation depends on providing our children with a complete education that includes music. -Gerald Ford